Complex analysis,maximal immersions and metric singularities

This paper introduces a complex analysis for the wave equation and for a singular second-order partial differential equation. As a main application of this complex analysis we construct type changing zero mean curvature immersions into Minkowski space. We also prove the existence of isothermal coordinates on a Lorentzian surface using this complex analysis and characterize flat maximal surfaces by their Gauss image. Finally we study the metric singularities of maximal immersions and semi Riemannian manifolds in general.     

A Bonnet theorem for isometric immersions into products of space forms

We prove a Bonnet theorem for isometric immersions of semi-Riemannian manifolds into products of semi-Riemannian space forms. Namely, we give necessary and sufficient conditions for the existence and uniqueness (up to an isometry of the ambient space) of an isometric immersion of a semi-Riemannian manifold into a product of semi-Riemannian space forms.     

Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.     

Decompositions of para-complex vector bundles and para-complex affine immersions

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ? over a para-complex manifold. First we obtain results on the connections induced on the subbundles, their second fundamental forms and their curvature tensors. In particular we analyze para-holomorphic decompositions. Then we introduce the notion of para-complex affine immersions and apply the above results to obtain existence and uniqueness theorems for para-complex affine immersions. This is a generalization of the results obtained by Abe and Kurosu [AK] to para-complex geometry. Further we prove that any connection with vanishing (0, 2)-curvature, with respect to the grading defined by the para-complex structure, induces a unique para-holomorphic structure.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

球面中具有平行平均曲率向量的2—调和等距浸入

本文讨论了单位球面中具平行平均曲率向量的２－调和等距浸入，获得其成为极小浸入的充分条件和关于第二基本形式的Ｐｉｎｃｈｉｎｇ定理．     

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN＇S INEQUALITY

A submanifold in a complex space form is called slant it it has constant Wirtinger angles. B, Y, Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP^2 and CH^2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen‘s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersion sin CP^n and CH^n with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen‘s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.     

Enclosure theorems for extremals of elliptic parametric functionals

In the following paper we study parametric functionals. First we introduce a generalized mean curvature (so called F-mean curvature). This enables us to describe extremals of parametric funcionals as surfaces of prescribed F-mean curvature. Furthermore we give a differential equation for arbitrary immersions generalizing and apply this equation to surfaces of vanishing and prescribed F-mean curvature. Especially we prove non-existence results for such surfaces generalizing Theorems by Hildebrandt and Dierkes [3], [6]. Received: 11 May 2001 / Accepted: 11 July 2001 / Published online: 12 October 2001     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Tightness of manifolds with H-spherical ends

In this article we prove a Chern–Lashof inequality for immersions of manifolds with H-spherical ends. Related to this inequality we discuss different types of tightness. In particular we shall prove that an immersion of a manifold with at least two H-spherical ends is tightly immersed only if the ends are of a certain geometric type (Wintgen immersion).     

The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type

In this paper, we define an equiaffine immersion of general codimension and the Lipschitz-Killing curvature for the immersion. Furthermore, we prove theorems of Gauss-Bonnet type and Chern-Lashof type for the immersion.     

Vanishing cohomology theorems and stability of complex analytic foliations

In this paper we deal with a complex analytic foliation of a compact complex manifold endowed with a bundle-like metric and give a transversally holomorphic rigidity theorem (Theorem 9.1) for these foliations, depending on curvature conditions. We give some examples for which we study holomorphic rigidity. The classical vanishing theorems of Nakano, Griffiths and Le Potier are the main tools we use to prove our results.     

Problema di Dirichlet per l’equazione delle superfici di curvatura media assegnata con dato infinito

Riassunto In questo lavoro si studia il problema di Dirichlet per l’equazione delle superfici di curvatura media assegnata con dato infinito. Si dimostra una condizione necessaria e sufficiente affinchè esista una soluzione. Si dimostra pure un teorema di unicità.

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Summary In this paper we consider the Dirichlet’s problem for surfaces of prescribed mean curvature with infinite data. We prove a necessary and sufficient condition for the existence of the solution. Finally we prove an uniqueness theorem.

     

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersions covered by connection-preserving equivariant immersions among the total spaces of the canonical circle bundles) thus relating the geometry of a pseudohermitian immersion from a strictly pseudoconvex CR manifold $M$ into an odd dimensional sphere, to the spectrum of the sublaplacian on $M$ .     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

具有非负Ricci曲率和次大体积增长的完备流形(英文)

本文研究了具有非负Ricci曲率和次大体积增长的完备黎曼流形的拓扑结构问题.利用Toponogov型比较定理及临界点理论,获得了流形具有有限拓扑型的结果,推广了H.Zhan和Z.Shen的定理,并且还证明了该流形的基本群是有限生成的.